[Paper Review] Probability Series Expansion Classifier that is Interpretable by Design

Sapan Agarwal, Corey Hudson|arXiv (Cornell University)|Oct 27, 2017

Adversarial Robustness in Machine Learning1 references3 citations

TL;DR

This paper proposes a novel, inherently interpretable classifier that estimates class probabilities directly from measured training data probabilities using a hierarchical series expansion. By refining estimates through multi-level feature combinations and weighting extreme probabilities more heavily, it achieves Random Forest-level accuracy while enabling full auditability of feature contributions and model behavior.

ABSTRACT

This work presents a new classifier that is specifically designed to be fully interpretable. This technique determines the probability of a class outcome, based directly on probability assignments measured from the training data. The accuracy of the predicted probability can be improved by measuring more probability estimates from the training data to create a series expansion that refines the predicted probability. We use this work to classify four standard datasets and achieve accuracies comparable to that of Random Forests. Because this technique is interpretable by design, it is capable of determining the combinations of features that contribute to a particular classification probability for individual cases as well as the weightings of each of combination of features.

Motivation & Objective

• Address the critical need for interpretable machine learning in high-stakes decision systems where model transparency is essential.
• Overcome the inherent opacity of complex models like Random Forests, which sacrifice interpretability for accuracy.
• Develop a classifier that directly uses measured training data probabilities to estimate class outcomes, avoiding complex optimization and regularization.
• Enable direct auditing of which feature combinations and individual features contribute most to a specific prediction.
• Maintain high predictive accuracy comparable to state-of-the-art models like Random Forests while ensuring interpretability by design.

Proposed method

• Estimate class probabilities using a hierarchical series expansion of measured probabilities from training data, starting from the most specific feature combinations.
• Refine probability estimates iteratively by combining lower-level probabilities (e.g., P(Y|xi) and P(Y|xi,xj)) into higher-level estimates using weighted averaging.
• Apply a scaling weight Si = 1 / (Pi′ × (1 − Pi′)) to emphasize probabilities near 0 or 1, ensuring strong evidence is given greater influence.
• Incorporate data count and uncertainty corrections via normalization terms (Ni / N) and utility weights (Ui) to reduce noise from low-sample estimates.
• Use Gini impurity or entropy to selectively prune less informative feature combinations, limiting model size without sacrificing accuracy.
• Handle missing features by omitting corresponding probability terms during estimation, preserving robustness in real-world scenarios.

Experimental results

Research questions

• RQ1 Can a classifier be designed from the ground up to be interpretable without sacrificing predictive accuracy?
• RQ2 How can measured training data probabilities be systematically combined to produce accurate and interpretable class predictions?
• RQ3 To what extent can feature-level and combination-level contributions to a prediction be quantified and visualized?
• RQ4 How do extreme probability estimates (near 0 or 1) affect overall prediction stability, and how can they be weighted appropriately?
• RQ5 Can a series expansion of probabilities achieve performance comparable to ensemble methods like Random Forests while remaining fully transparent?

Key findings

• The Probability Series Expansion Classifier achieves classification accuracy comparable to scikit-learn’s Random Forest implementation across four standard datasets.
• The model enables full interpretability by directly identifying which feature combinations contributed most to a prediction, with explicit contribution weights.
• In a misclassified instance from the congressional voting records dataset, feature combinations such as x3=1, x9=1, x15=0 contributed 12.5% weight with 100% probability of Republican classification.
• Individual features like x9=1 and x3=1 were assigned high weights (17% and 16%) due to their strong predictive power in specific combinations.
• The model successfully identified contradictory evidence in the training data, such as feature combinations predicting both 0% and 100% Republican affiliation.
• The inclusion of data count and uncertainty corrections significantly improved robustness, preventing blowup in scaling weights from low-sample estimates.